A local strategy to decide the Alperin and Dade conjectures (Q1270405)

A new local strategy is presented which exploits both the maximal and \(p\)-local subgroup structure of a given simple group in order to decide the Alperin and Dade conjectures for this group. The main result of this paper is the verification of these conjectures for the Conway group \(Co_2\). Let \(G\) be a finite group, \(p\) a prime, and \(B\) a \(p\)-block of \(G\). Alperin conjectured that the number of \(B\)-weights equals the number of irreducible Brauer characters of \(B\). A chain \(C\): \(U_0\subset U_1\subset\cdots\subset U_m\) of \(p\)-subgroups of a finite group \(G\) is called radical if \(U_0=O_p(G)\) and \(U_i=O_p(\bigcap_{j=0}^i N_G(U_j))\), \(1\leq i\leq m\). For such a chain \(C\) we denote \(\bigcap_{j=0}^i N_G(U_j)\) by \(N_G(C)\) and by \(| C|\) we denote the length \(m\) of \(C\). Moreover, for a given \(p\)-block \(B\) of \(G\) and a non-negative integer \(d\), let \(k(N_G(C),B,d)\) denote the number of irreducible characters \(\varphi\) of \(N_G(C)\) inducing up to \(B\) and such that \(p^d\) is the highest power of \(p\) dividing \(| N_G(C)|/\varphi(1)\), that is \(\varphi\) has defect \(d\). Note that if two radical chains \(C\) and \(C'\) are \(G\)-conjugate then \(k(N_G(C),B,d)=k(N_G(C'),B,d)\) for all \(B\) and \(d\). Dade made the following Ordinary Conjecture. Suppose that the finite group \(G\) satisfies \(O_p(G)=1\) and that the \(p\)-block \(B\) of \(G\) is not of defect \(0\). Then \(\sum'(-1)^{| C|}k(N_G(C),B,d)=0\) for all \(d\geq 0\), where \(\sum'\) denotes the sum over a set of representatives of the \(G\)-conjugacy classes of radical chains of \(G\).